Finite Group Databasebeta

← #509572

Subgroups — #509572

Order 512.

541 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2243C211yesmaximal, characteristic
2C2×#2226C211yesmaximal, characteristic
3#28813C211yesmaximal, characteristic, complemented
4#26252C211yesmaximal, characteristic
5#13531C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic
7#13475C211yesmaximal, characteristic, complemented
8#13452C211yesmaximal, characteristic, complemented
9#13408C211yesmaximal, characteristic
10#13364C211yesmaximal, characteristic, complemented
11#13168C211yesmaximal, characteristic, complemented
12#13132C211yesmaximal, characteristic, complemented
13#13130C211yesmaximal, characteristic
14#13105C211yesmaximal, characteristic, complemented
15#12937C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2551—21no—
21#2447—21no—
22#2243—24no—
23#2226—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#346C2211yescharacteristic
38#2057—21no—
39#2057—21no—
40#2055—21no—
41#2042—21no—
42#1970C2211yescharacteristic
43#1969C2211yescharacteristic
44#1969C2211yescharacteristic
45#1947—21no—
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1877—21no—
50#1845C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—41no—
67C2×#124—21no—
68C2×C4×D4—21no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#500—22no—
74#500—22no—
75#500—41no—
76#500—41no—
77#495—22no—
78#495—41no—
79C4×D8—22no—
80C4×D8—41no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×#103—21no—
92C2×#103—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#397—41no—
99#397—41no—
100#395—41no—
101#395—41no—
102#391—21no—
103#391—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—41no—
109#385—41no—
110#384—21no—
111#381—41no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D411yescharacteristic
114#367—22no—
115#367—22no—
116#367—41no—
117#366—22no—
118C4⋊C16—22no—
119#358—22no—
120#358—22no—
121#358—22no—
122#358—41no—
123#357—22no—
124#357—22no—
125#357—22no—
126#357D414yes—
127#346—22no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—21no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×D4—21no—
138C23×C4—41no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×SD16—42no—
146C2×SD16—42no—
147C2×SD16—81no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8C2×D412yes—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2411yescharacteristic
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—81no—
161C4⋊Q8—81no—
162C4⋊Q8—84no—
163#124—42no—
164#124—81no—
165C4×D4—42no—
166C4×D4—81no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4D811yescharacteristic
181C2×C4⋊C4D811yescharacteristic
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C42—41no—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16C2×D412yes—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—42no—
214#107—42no—
215#107—81no—
216#107C2×D412yes—
217#107#4112yes—
218C4⋊C8—22no—
219#103—81no—
220#103—81no—
221#102—42no—
222#102—81no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95D811yescharacteristic
238#95D811yescharacteristic
239C24—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—82no—
247C2×Q8—82no—
248C2×Q8—82no—
249C2×Q8—82no—
250C2×Q8—161no—
251C2×Q8—161no—
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—161no—
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C22×D411yescharacteristic
275Q16—41no—
276Q16—41no—
277Q16—41no—
278Q16—41no—
279SD16—84no—
280D8—22no—
281D8—22no—
282D8—41no—
283D8—41no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8#13712yes—
293C2×C8C22×D411yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8H(5,𝔽2)11yescharacteristic
296C2×C82−1+411yescharacteristic
297C2×C82−1+411yescharacteristic
298C2×C82−1+411yescharacteristic
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—161no—
319C4⋊C4—161no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—161no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—82no—
332C16—22no—
333C16—41no—
334C16—41no—
335C16—41no—
336C23—42no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23C22×D811yescharacteristic
344Q8—81no—
345Q8—81no—
346Q8—81no—
347Q8—81no—
348Q8—164no—
349Q8—164no—
350D4—42no—
351D4—42no—
352D4—81no—
353D4—81no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—164no—
392C2×C4#50912yes—
393C2×C4#51012yes—
394C2×C4#53611yescharacteristic
395C2×C4D4211yescharacteristic
396C2×C4#54811yescharacteristic
397C2×C4C2×#13711yescharacteristic
398C2×C4#57611yescharacteristic
399C2×C4#57711yescharacteristic
400C8—22no—
401C8—41no—
402C8—41no—
403C8#53611yescharacteristic
404C8#54211yescharacteristic
405C8#54711yescharacteristic
406C8#56311yescharacteristic
407C22—82no—
408C22—82no—
409C22—82no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22#322611yescharacteristic
415C22D4×D811yescharacteristic
416C22#335811yescharacteristic
417C22C2×#50911yescharacteristic
418C22C2×#51011yescharacteristic
419C22#341611yescharacteristic
420C22#341611yescharacteristic
421C4—22no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—82no—
428C4—82no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4#267411yescharacteristic
435C4#324111yescharacteristic
436C4#326611yescharacteristic
437C4#337511yescharacteristic
438C2—162no—
439C2#3080511yescharacteristic
440C2#3380411yescharacteristic
441C2#3381011yescharacteristic
442C2#3383211yescharacteristic
443C2#3383811yescharacteristic
444C2#3393611yescharacteristic
445C2#3394011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #28813C2semidirect
5. #13531C2semidirect
7. #13475C2semidirect
8. #13452C2semidirect
10. #13364C2semidirect
11. #13168C2semidirect
12. #13132C2semidirect
14. #13105C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#22432
C2×#22262
#288132
#262522
#135312
#134802
#134752
#134522
#134082
#133642
#131682
#131322
#131302
#131052
#129372